Factorization

Basic Algebra > Factorization

Consider the following example:
12 = 3 × 4
i.e., 12 is product of 3 and 4.
3 and 4 are called factors or divisors of 12
12 is also equal to 2 × 6.

Here again, numbers 2 and 6 are called factors or divisors of 12.
The process of writing number 12 into product of 3 and 4 or 2 and 6 is called factorization.

Which is the other way of factorizing 12?
It is 12 = 12 × 1
Above is an example of factorizing a monomial (here 12) into three pairs of other monomials 3, 4 and 2, 6 and 1,12.

An algebraic expression can also be factored into one or more factors.
For example, the polynomial x2 + 2x can be written as product of two binomials namely, x and (x+2) x and (x+2) are called factors or divisors of x and (x+2) .
Also, they are polynomials, while, x is monomial and x+2 is a binomial.

So, what is Factorization?

Factorization

Expressing polynomials as product of other polynomials that cannot be further factorized is called Factorization

We will discuss all the various types of factorization in this lesson.
They are standard types found across all books on algebra. Here, we go.

1. First Type: Factorization into Monomials:

Remember distributive property? It is
a (b + c ) = ab + ac

Example 1: Factorize ax + bx

Solution:
In ax and bx, x is common and also a common factor.
Write the common factor x in the polynomial ax + bx outside as x(a + b)
Now, both x and a + b are factors of the polynomial
ax + bx
So, factorization of
ax + bx = x(a + b)
Here the common factor x is a monomial.

Important Note:

In 10(a + b), 10 is one of the two factors and we do not factorize 10. We need not write 10 as 1×10 or 2×5. Factors that are numbers are not further factorized.

What is Complete Factorization?

Example: Factorize: 12x2 + 18x3

Solution:12x2 + 18x3 can be factorized into any of the following ways:
12x2 + 18x3 = x (12x +18x2)
12x2 + 18x3 = 6x (2x + 3x2)
12x2 + 18 x3 = 6x2 (2 + 3x)
There may be more ways of expressing the polynomial 12x2 + 18x3as product of two factors. Let us limit to the above three ways.

Which of the above three ways is accepted as standard form of factorization?

The third one.
In the third one, 6x2is the greatest common divisor orhighest common factor of the two algebraic expressions (monomials)12x2 and 18x3

Complete Factorization

In complete factorization, the greatest common factor is written as the common factor

Example 1: Factorize x2y2 + y2

Solution:
Completely factorize x2y2 + y2, i.e. write the greatest common factor of x2y2and y2as the common factor. y2is the highest common factor. To find the other factor, divide each term by the H.C.F. and add the quotients.
(x2y2)/y2 = x2 and y2/y2 = 1
Sum of the quotients is x2 + 1, which is the other factor. Therefore,
x2y2 + y2 = y2(x2 + 1)

2. Second type of Factorization: By Grouping of Terms

Those terms which yield a common factor on grouping!
Group the terms like this: a2 + ab + bc + ac
a is the common factor in a2 + ab and b is the common factor in ab + bc
a2 + ab = a (a + b) and bc + ac = c (b + a)
now, the common factor is the binomial (a + b).

{note that a + b is same as b + a, i.e., order of addition does not matter, since addition follows closure property which states a + b = b + a}

Solution:
Group terms with similar literal coefficients and numerical coefficients.
One group is ax + bx, in which x is the same literal coefficient,
and the other is ay + by, in which the y is the same literal coefficient,
In ax + bx, the common factor is x and in
ay +by, the common factor is y.
on factorization with common factors, we have:
ax + bx = x ( a + b ) and
ay + by = y (a + b)
so ax + bx + ay + by = x (a + b) + y (a + b). (a + b) is the common binomial factor for the next step of factorization: (a + b)(x + y)
Factorization of ax + bx + ay + by = (a + b)(x + y)